Summary: A Full Characterization of Quantum Advice
Scott Aaronson
MIT
Andrew Drucker
MIT
Abstract
We prove the following surprising result: given any quantum state on n qubits, there exists
a local Hamiltonian H on poly (n) qubits (e.g., a sum of two-qubit interactions), such that any
ground state of H can be used to simulate on all quantum circuits of fixed polynomial size.
In terms of complexity classes, this implies that BQP/qpoly QMA/poly, which supersedes the
previous result of Aaronson that BQP/qpoly PP/poly. Indeed, we can exactly characterize
quantum advice, as equivalent in power to untrusted quantum advice combined with trusted
classical advice.
Proving our main result requires combining a large number of previous tools--including a
result of Alon et al. on learning of real-valued concept classes, a result of Aaronson on the learn-
ability of quantum states, and a result of Aharonov and Regev on `QMA+ super-verifiers'--and
also creating some new ones. The main new tool is a so-called majority-certificates lemma,
which is closely related to boosting in machine learning, and which seems likely to find inde-
pendent applications. In its simplest version, this lemma says the following. Given any set
S of Boolean functions on n variables, any function f S can be expressed as the pointwise